Pasch's theorem
In geometry, Pasch's theorem, stated in 1882 by the German mathematician Moritz Pasch,[1] is a result of Euclidean geometry which cannot be derived from Euclid's postulates.
The statement is as follows. Given points a, b, c, and d on a line, if it is known that the points are ordered as (a, b, c) and (b, c, d), then it is also true that (a, b, d).[2] [Here, for example, (a, b, c) means that point b lies between points a and c.]
See also
Notes
- ↑ Vorlesungen über neuere Geometrie (Leipzig, 1882)
- ↑ Coxeter (1969, p. 179) states the result in 12.274 but does not refer to it specifically as Pasch's theorem.
References
- Coxeter, H.S.M. (1969). Introduction to geometry (2nd ed.). John Wiley and Sons. ISBN 0-471-18283-4. Zbl 0181.48101.